Shell Method (Volumes of Revolution)
This construction is intended to allow exploration of the "Shell Method" (or "Shells") for calculating volumes of revolution. This is typically a Calculus I or Calculus II topic in college. Shells is not in the high school AP Calculus AB/BC curriculum although a motivated student may find it accessible.
The integral formula for the Shell Method can be found in any calculus textbook or comprehensive internet calc resource, and I won't present it here.
The Disk/Washer method is typically introduced first when learning volumes of revolution. When learning that method, a student should understand that the differential dx (or dy) may conceptually represent the infinitesimal thicknesses of infinitely many disks or washers stacked upon each other.
In a similar manner, for Shells the differential dx (or dy) may conceptually represent the infinitesimal thicknesses of a infinitely many right-circular cylindrical shells nested inside each other.
Perhaps an ambitious student will develop the formula on their own. Or even if the formula is looked up elsewhere, perhaps this construction will help illuminate why the formula is what it is.
Controls in the GeoGebra construction below should become apparent with a little tinkering around. But here are some things that may not be so obvious:
- In addition to editing the equations for f(x) and g(x), you may drag their graphs (i.e. translate in the x-y plane) dynamically in the 2D graphics view.
- In the 2D graphics view: In addition to dragging the axis of revolution or points A or B dynamically, you may also just select the axis or point and then use ←→ arrow keys on a keyboard to nudge left/right in increments of 0.1. Holding shift key while using arrows changes increment to 0.01. Holding ctrl key while using arrows changes increment to 1.
- In the 3D graphics view: Click/drag to rotate view to a different angle. On a keyboard, hold shift key while click/dragging to pan (or translate) the view. Holding shift key and just clicking will toggle the option to pan in the x-y plane or along the z-axis.